Physical Predictions in Closed Quantum Gravity

Abstract

Recent developments in gravitational path integrals indicate that the nonperturbative physical Hilbert space of a closed universe is one-dimensional within each superselection sector. This raises a basic puzzle: how can a unique quantum-gravity state give rise to semiclassical physics, measurement outcomes, and classical probabilities? In this paper, we develop a framework in which nontrivial and statistically stable predictions emerge despite the one-dimensionality of the fully constrained Hilbert space. The key idea is to extract physical predictions in an enlarged, unconstrained Hilbert space by conditioning on observational data. We show that partial observability -- reflecting the limited access of observers to the degrees of freedom of the universe -- suppresses ensemble fluctuations associated with microscopic structure in the gravitational path integral, thereby restoring semiclassical predictability with exponential accuracy. We formulate the construction explicitly including contributions from the Hartle--Hawking no-boundary state, define a gauge-invariant Hilbert space for observations via a density operator, and generalize the formalism to conditioning on histories, clarifying the emergence of classical probabilities and an effective arrow of time. Finally, we explore whether this framework can support a realistic cosmology and identify assumptions that the underlying theory of quantum gravity must satisfy.

Figures (9)

• Figure 1: Representative gravitational path-integral geometries contributing to the probability of obtaining the measurement outcome $X_i$. Green circles and blue dots represent yarmulke and crotch singularities, respectively.
• Figure 2: Examples of conditioning geometries containing disconnected universes. Disconnected components on the conditioning hypersurface that carry no observational data are traced out.
• Figure 3: Saddles contributing to the gravitational path integral for $P(X_i)^2$. The first line shows contributions corresponding to the square of the saddles for $P(X_i)$, while the second line gives contributions that do not have corresponding contributions in the computation of $P(X_i)$.
• Figure 4: Saddles contributing to the gravitational path integral for $P(X_i)^2$. All three have equal magnitude, but only the first equals the square of the saddle for $P(X_i)$.
• Figure 5: Saddles contributing to the gravitational path integral for $P(X_i)^2$ when conditioning is restricted to a subset of the whole degrees of freedom. The topologies of the first and other diagrams differ. As a result, $P(X_i)^2$ is dominated by the first saddle, suppressing replica-wormhole contributions.